Question

The sides of the triangle are in the ratio of \[5:12:13\] and its perimeter is \[450m\], find its area.

Answer 1

Hint: Here, in the question, we have been given the ratio of the three sides of the triangle and its perimeter and we are asked to find the area of the same triangle. We will, at first, use the perimeter formula to calculate the sides of the triangle and then we will find the area using the Heron’s Formula.
Formula used:
Perimeter of triangle=Sum of all three sides of a triangle
Area of triangle= \[\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where,
\[a,b,c\] are the sides of the triangle
\[s = \dfrac{{a + b + c}}{2}or\dfrac{\text{Perimeter}}{2}\]

Complete step-by-step solution:
Given, Perimeter of triangle= \[450m\]
Ratio of three sides of the triangle= \[5:12:13\]
Let the sides of the triangle be \[5x\],\[12x\] and \[13x\]

Perimeter of triangle=Sum of all three sides of a triangle
\[ \Rightarrow 450 = 5x + 12x + 13x\]
Simplifying it, we get,
\[ 30x = 450 \\
   \Rightarrow x = 15 \]
Putting the value of \[x\] in the sides of the triangle, we get,
Three sides of the triangle are \[75m\],\[180m\] and \[195m\].
Now, to calculate the area of the triangle, we will use Heron’s formula.
Area of triangle=\[\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \] , where \[a,b,c\] are the sides of the triangle
\[s = \dfrac{{a + b + c}}{2}\] or \[\dfrac{\text{Perimeter}}{2}\]
\[ \Rightarrow s = \dfrac{{450}}{2}\]
\[ \Rightarrow s = 225\]
Therefore, Area of triangle=\[\sqrt {225\left( {225 - 75} \right)\left( {225 - 180} \right)\left( {225 - 195} \right)} \]
Area of triangle \[ = \sqrt {225 \times 150 \times 45 \times 30} \]
Or we can write it as,
Area of triangle \[ = \sqrt {225 \times 15 \times 10 \times 15 \times 3 \times 10 \times 3} \]
Taking squares outside the root, we get,
Area of triangle \[ = 15 \times 15 \times 10 \times 3\]
Area of triangle= \[6750{m^2}\] \[\]

Note: Alternatively, if we observe the ratio of the sides given, it is clearly visible that the given triangle is right angled triangle as all the sides satisfies Pythagoras theorem which states that the square of length of hypotenuse (the longest side) will be equal to the squares of lengths of other two sides or \[{H^2} = {P^2} + {B^2}\], where \[H\] is the hypotenuse, \[P\] is the perpendicular and \[B\] is the base of the triangle. In this case, we could have find the area using the formula, Area of triangle= \[\dfrac{1}{2} \times base \times height\]